Theorem frgrncvvdeqlem1 30331

Description: Lemma 1 for frgrncvvdeq 30341. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem1	⊢ (𝜑 → 𝑋 ∉ 𝑁)

Proof of Theorem frgrncvvdeqlem1

Step	Hyp	Ref	Expression
1		frgrncvvdeq.xy	. . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
2		df-nel 3053	. . . . 5 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷)
3		frgrncvvdeq.nx	. . . . . 6 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4	3	eleq2i 2836	. . . . 5 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
5	2, 4	xchbinx 334	. . . 4 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
6	1, 5	sylib 218	. . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7		nbgrsym 29398	. . 3 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
8	6, 7	sylnibr 329	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9		frgrncvvdeq.ny	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
10		neleq2 3059	. . . 4 ⊢ (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
11	9, 10	ax-mp 5	. . 3 ⊢ (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12		df-nel 3053	. . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
13	11, 12	bitri 275	. 2 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
14	8, 13	sylibr 234	1 ⊢ (𝜑 → 𝑋 ∉ 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  {cpr 4650   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082   NeighbVtx cnbgr 29367   FriendGraph cfrgr 30290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-nbgr 29368

This theorem is referenced by:  frgrncvvdeqlem7  30337  frgrncvvdeqlem8  30338  frgrncvvdeqlem9  30339